Waseda Young Researchers Meeting on Geometry 2025

Abstract: Killing spinor is one of the special spinors. Existence of Killing spinor impose strong restrictions on the geometry of the underlying manifold, for example, Einstein condition. In this talk, we introduce a concept called Ricci Killing spinor which is generalization of Killing spinor on non-Einstein manifolds. Ricci Killing spinors are related to hypersurface theory. More precisely the existence of Ricci Killing spinor is “equivalent” to the manifold being embeddable into a Ricci-flat manifold as a hypersurface. Moreover the mean curvature of the hypersurface is equal to the Ricci curvature. 

Abstract: Every homogeneous Riemannian manifold of negative curvature is known to be isometric to a Lie group with a left invariant metric. We define an SNC-algebra to be a Lie algebra which admits an inner product of strictly negative curvature. Following the Heintze’s criteria for a Lie algebra to be an SNC-algebra, we are maked a complete classification for all SNC-algebras in dimension four. As a result, four dimensional SNC-algebras were classified into seven classes each of which contains some parameters. This take is based on joint work with Toru Goto.

Abstract: See here.

Abstract: Gromov introduced a notion of negative curvature for general metric spaces, not necessarily Riemannian manifolds. Metric spaces with such negative curvature are called Gromov hyperbolic spaces, and groups acting properly and cocompactly on them are called hyperbolic groups. Hyperbolic groups have been extensively studied and are known to have a wide range of algebraic applications. This raises the question: how can we formulate non-positive curvature for more general metric spaces or groups? One answer was given by Tomohiro Fukaya and Shin-ichi Oguni, who introduced the notion of coarsely convex spaces and coarsely convex groups. This class of spaces has a lot of favorable properties; for instance, analogues of the Cartan–Hadamard theorem and the coarse Baum–Connes conjecture hold. In this work, we study a generalization of coarsely convex groups called relatively coarsely convex groups. Roughly speaking, such a group becomes coarsely convex after breaking the structure of a certain family of subgroups. We define relatively coarsely convex groups and show that this property is preserved under certain amalgamated products and HNN extensions. This talk is based on joint work with Tomohiro Fukaya and Eduardo Martínez-Pedroza (arXiv:2503.08995).